Under CMS->Admin->Config->Manage Websites-> choose specific website (http://mysite/cms/CMS/Admin/SiteInformationList.aspx then choosing appropriate website)

One can configure multiple Host names, for example:

• localhost
• *
• mysite
• test.company.org
• and so on...

There is also section called URL, this is the only place where one can configure protocol, ie.: http://mysite or https://mysite

If http://mysite configured then following request works:

http://mysite/cms/cms/Stores/sitestructure/?siteUrl=http%3A%2F%2Fmysite%2F&allLanguages=true&dojo.preventCache=1401099189822

and this does not:

https://mysite/cms/cms/Stores/sitestructure/?siteUrl=https%3A%2F%2Fmysite%2F&allLanguages=true&dojo.preventCache=1401099189822

giving exception:

[ArgumentException: The specified url https://mysite/ does not match any site.]
   EPiServer.Cms.Shell.UI.Rest.SiteStructureStore.GetLanguageModelsForSite(String siteUrl, Boolean showAllLanguages) +1126
   EPiServer.Cms.Shell.UI.Rest.SiteStructureStore.Get(String siteUrl, Nullable`1 allLanguages) +221
   lambda_method(Closure , ControllerBase , Object[] ) +172
   System.Web.Mvc.ReflectedActionDescriptor.Execute(ControllerContext controllerContext, IDictionary`2 parameters) +261
   System.Web.Mvc.ControllerActionInvoker.InvokeActionMethod(ControllerContext controllerContext, ActionDescriptor actionDescriptor, IDictionary`2 parameters) +39
   System.Web.Mvc.<>c__DisplayClass13.<InvokeActionMethodWithFilters>b__10() +124
   System.Web.Mvc.ControllerActionInvoker.InvokeActionMethodFilter(IActionFilter filter, ActionExecutingContext preContext, Func`1 continuation) +855266
   System.Web.Mvc.ControllerActionInvoker.InvokeActionMethodFilter(IActionFilter filter, ActionExecutingContext preContext, Func`1 continuation) +855266
   System.Web.Mvc.ControllerActionInvoker.InvokeActionMethodWithFilters(ControllerContext controllerContext, IList`1 filters, ActionDescriptor actionDescriptor, IDictionary`2 parameters) +309
   System.Web.Mvc.ControllerActionInvoker.InvokeAction(ControllerContext controllerContext, String actionName) +855200
   EPiServer.Shell.Services.Rest.RestControllerBase.ExecuteCore() +301
   System.Web.Mvc.ControllerBase.Execute(RequestContext requestContext) +334
   EPiServer.Shell.Services.Rest.RestHttpHandler.ProcessRequest(HttpContextBase httpContext) +274
   System.Web.CallHandlerExecutionStep.System.Web.HttpApplication.IExecutionStep.Execute() +913
   System.Web.HttpApplication.ExecuteStep(IExecutionStep step, Boolean& completedSynchronously) +165

and once you configure 'CMS->Admin->Config->Manage Websites-> choose specific website' URL to https://mysite, then https version works properly and http one throws the same exception

and its even worse as one usualy have multiple hosts records in 'CMS->Admin->Config->Manage Websites-> choose specific website'

this does not work as well(while http://mysite configured in URL):

http://localhost/cms/cms/Stores/sitestructure/?siteUrl=http//localhost/&allLanguages=true&dojo.preventCache=1401099189823

Any pointers?

EPiServer 7.6.4 is used

PS. in all other aspects and other requests seems that protocol change and hostname change works properly.